First we discuss the case that \(M\)  is a homology sphere. Then we can get any overtwisted contact structure on \(M\)  from a fixed contact structure on \(M\)  by connected summing with overtwisted contact structures of \(S^3\)  [1]}. It follows that any two overtwisted contact structures on \(M\)  can be connected in \(\Gamma _M\) . Now let \(\xi _{\textrm {tight}}\)  be some tight contact structure on \(M\) . Then there there exists a single contact \((+1)\) -surgery along a Legendrian unknot in \((M,\xi _{\textrm {tight}})\)  yielding \((M,\xi _{\textrm {tight}})\# (S^3,\xi _1)\)  which is overtwisted and thus it follows that \(\Gamma _M\)  is connected.
If the underlying manifold is not a homology sphere it gets slightly more complicated since the classification of tangential 2-plane fields is more involved. As a further invariant we have the \(spin^c\)  structure of a contact structure. However, it is known that for a given contact structure with \(spin^c\)  structure \(\mathfrak {s}\)  we can get any other overtwisted contact structure with the same \(spin^c\)  structure \(\mathfrak {s}\)  by connected summing with the overtwisted contact structures on \(S^3\)  [1]}. Thus, we can apply the same argument as in the homology sphere case to deduce that \(\Gamma _{(M,\mathfrak {s})}\)  is connected.
It remains to show that there is no edge connecting two different \(spin^c\)  structures on \(M\) . For that we use Gompf's \(\Gamma \) -invariant which classifies \(spin^c\)  structures [3]}. Let \(\mathfrak {s}\)  be a \(spin^c\)  structure on \(M\)  and \(\xi \)  be a contact structure inducing \(\mathfrak {s}\) . Let \(L_0\)  in \((M,\xi )\)  be a Legendrian knot such that contact \((+1)\) - or contact \((-1)\) -surgery along \(L_0\)  yields another contact structure \(\xi ^{\prime }\)  on \(M\) . We want to show that \(\xi ^{\prime }\)  induces the same \(spin^c\)  structure \(\mathfrak {s}\) . For that we choose a spin structure \(\mathfrak {t}\) , describe \((M,\xi )\)  by a contact surgery diagram along a Legendrian link \(L=L_1\cup \ldots \cup L_n\)  in \((S^3,\xi _{\mathrm {st}})\) , and present \(L_0\)  as a knot in the exterior of \(L\) . To show that \(\xi \)  and \(\xi ^{\prime }\)  induce the same \(spin^c\)  structures it is enough to compute that Gompf's \(\Gamma \)  invariants of \(\xi \)  and \(\xi ^{\prime }\)  with respect to \(\mathfrak {t}\)  agree. We present \(\mathfrak {t}\)  via a characteristic sublink \((L_j)_{j\in J}\) , \(J\subset \lbrace 1,\ldots ,n\rbrace \) , of \(L\) . Since the homologies of \(M\)  and \(L_0(\pm 1)\)  agree we deduce that \(\mu _{L_0}\)  is nullhomologous in \(L_0(\pm 1)\)  and that \(J\)  is also a characteristic sublink of the surgery diagram \(L_0\cup L\)  of \(L_0(\pm 1)\) . Then we can use the formula for computing Gompf's \(\Gamma \) -invariant from [4]} to compute
\(\Gamma (\xi ^{\prime },\mathfrak {t})-\Gamma (\xi ,\mathfrak {t})=\frac{1}{2} \left(\sum _{i=0}^n r_i\mu _i+\sum _{j\in J}(Q^{\prime }\mu )_j\right)-\frac{1}{2} \left(\sum _{i=1}^n r_i\mu _i+\sum _{j\in J}(Q\mu )_j\right)=0,\) 

where \(r_i\)  denotes the rotation number and \(\mu _i\)  the meridian of \(L_i\) , \(Q^{\prime }\)  and \(Q\)  denotes the linking matrices of \(L_0\cup L\)  and \(L\) , and all non-canceling terms are multiples of \(\mu _0=0\) .
